MinT - Best Known (68, 68+83, s)-Nets in Base 3

Best Known (68, 68+83, s)-Nets in Base 3

(68, 68+83, 48)-Net over F₃ — Constructive and digital

Digital (68, 151, 48)-net over F₃, using

t-expansion [i] based on digital (45, 151, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(68, 68+83, 72)-Net over F₃ — Digital

Digital (68, 151, 72)-net over F₃, using

t-expansion [i] based on digital (67, 151, 72)-net over F₃, using
- net from sequence [i] based on digital (67, 71)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(68, 68+83, 410)-Net in Base 3 — Upper bound on s

There is no (68, 151, 411)-net in base 3, because

1 times m-reduction [i] would yield (68, 150, 411)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 399497 837381 270341 125769 351075 891244 333385 166684 227480 022408 639102 909095 > 3¹⁵⁰ [i]